Category Theory

For the last five years mathematics has been my passion, as well as my main focus in life. This passion for mathematics will hopefully not diminish, as I am now heading into four more years of studies and research through a PhD in mathematics at NTNU. I am joining a project, called Tensor triangulated geometry in Trondheim, so today I thought I would explore the definition of one of the main players in this theory, namely tensor triangulated categories. We have already met half of this structure during our several encounters of monoidal categories, but the other half remains, as well as how to glue them together into a cohesive joint structure. ...

For those that don’t know I am a fifth year mathematics student at NTNU, meaning I am finishing my masters degree after this semester. During my time at NTNU I have had some wonderful classes, and some wonderful teachers. Since most I post about on this blog is related to topology, it is very safe to assume that some of my most memorable courses are exactly the topology courses. I very recently looked at my notes from my first topology course, or rather one of the two first, as I took two in parallel during my fourth semester. The course was focused on differential topology and the study of smooth manifolds. It was taught by my now supervisor, but on a couple of the last lectures we had some guest appearances from the other topology professors at NTNU. One of these guest lectures is the focus of todays blog post. ...

Since one of my main mathematical interests is homotopy theory, im bound to often bump into things that require the use of base-points. This has long been the classical way to study spaces, especially in terms of homotopy groups. When I was introduced to these so-called pointed spaces, I couldn’t help but feel that these we less natural, or more ad hoc, than regular spaces. I didn’t know much about categories then, but have since learned it is usually in this context that some form of naturality occur. It turns out that pointed spaces actually come from a very nice natural categorical construction, which of course is the focus of this post. I will assume introductory knowledge of categories, and I will try to keep this short for once. ...

This post is part two of a little two-part miniseries about defining the cosmos. To learn what a cosmos is in mathematics, or rather what we want it to be, you can read the first part. There we described a cosmos as a nice place to enrich a category, or a nice place to do enriched category theory, and to quickly recap, an enriched category is a category where we have objects of morphisms instead of just a collection of them, and these objects come from some monoidal category. In this post we will continue the story, and focus more on the definition rather than the setup. ...

I think there are many parts of physics worth studying for mathematicians, and the physical notion of a cosmos may be one of them, but, this post is not about physics. Even though the usual field of study one thinks of when hearing the word “cosmos” is physics, there is also a type of mathematical object with the same name. This type of object does have that name for a reason, which is not clear maybe from the object it self, but from what one can do with and in such an object. In physics, the cosmos is a word for the universe, but it includes also the universes structure. The cosmos is not just “all the stuff that exists”, but also their relations and complex interactions. It is therefore kind of the background, or the playing field of physics. ...

A part of mathematics I really an starting to enjoy more is mathematics that explain or develop connections between geometry or topology, and algebra. The first two posts on this blog was focused on developing some geometrical insight to two lemmas from commutative algebra, namely Noether’s normalization lemma and Zariski’s lemma. There are many more such connections worth discussing and exploring, and today I want to focus on one of these “bridges” between geometry and algebra, namely Swan’s theorem. This theorem tells us how nice objects “over” another object in geometry relate to nice objects “over” another object in algebra. ...

This is part 7 of a series leading up to and exploring model categories. For the other parts see the series overview. Finally we have made it to the destination we set, namely, more abstraction. This post is focused on the definition and intuition on model categories, which abstracts the objects we have been studying for some weeks, namely fibrations and cofibrations. The main definition is that of a model structure on a category, which together with a nice category will form the definition of a model category. So, why do we want this? There are more than one reason. ...